My last post on my quibbles about closure began by saying I’d explain why the issue isn’t merely terminological. Apparently, at 5 a.m., I forget by the end of the post what I said at the beginning…

Anyway, here’s the explanation.

The fundamental objection to widening the scope of what counts as a closure principle is that it makes a mockery of the debate about closure, making deniers of closure into defenders of such an implausible claim that no one would have thought to deny it. Here’s the principle critics of closure would have to deny:

there is some true epistemic principle that says that if take a set of propositions that are known by some person, there is some operation or process involving the deductive consequence relation in some way which is such that, when applied so as to produce some proposition q, implies that q falls into a favorable epistemic category (perhaps involving the concept of knowledge) for that person.

The favorable epistemic category might have to do with being in a position to know, which some would find unproblematic, but it might equally have to do with being justified in the way required for knowledge. To require of closure deniers the intention to undermine this principle is to raise the bar on their objections to such a height that it makes a mockery of the dispute. The proper stance to take on the dispute requires honoring enough of the mathematical home of the concept of closure so that the argument is about whether, in some way, an operation or process on a set of propositions having a certain property yields a result having that same property. Taken in this sense, there is an interesting dispute about whether there are any correct closure principles in epistemology; taken more broadly as in the last principle above, there has not been any such dispute.

On the other side of the fence here, it is true that critics of closure have given arguments that threaten both closure and non-closure epistemic principles. For example, Dretske’s disguised mule case threatens not only closure principles but also non-closure principles about being in a position to know that the object is not a cleverly disguised mule given that one knows that it is a zebra. People who’ve rejected the Dretske argument usually do so by citing some non-closure principle that it undermines, claiming that giving up that principle is just too much (the exception to this claim is John Hawthorne’s rejection in the Blackwell *Debates* volume). If they want to turn this response into a defense of closure, they need to do what Hawthorne attempts to do: formulate a closure principle that survives the example.

As per my comment in the previous post, I do not see that there is a need to broaden the notion of epistemic closure principle provided you include the right epistemic operators for being in a position to know, being justified, believing, knowing, and so on. Maybe I’m wrong, but I think you are being restrictive on what sorts of epistemic operators are permissible.

Closure isn’t a matter of what battery of operators one has available. That’s an issue for which epistemic principles are expressible in whatever formal language one chooses to use. Closure principles require a specification of something that is closed, and when a thing is closed (under some operation or function), we’ll have an equality or identity between what we began with and what we end up with.

_Closure principles require a specification of something that is closed, and when a thing is closed (under some operation or function), weâ??ll have an equality or identity between what we began with and what we end up with_.

Not sure why you to put it this way. Suppose, for instance, that S = {a, b}. Now suppose we close S under disjunction. S’={a,b,(avb),(bva),((avb)v(avb)), . . .}. The closure of S doesn’t take S into S. That is it doesn’t map S on to members already in S. So it is unusual to say that the set we performed the function on (viz. S) is identical to the set closed under disjunction (viz. S’). You must mean something else. But I’m not sure what, exactly.

Mike, to say that we close a set under an operation is not to say that the original set was closed under that operation. Otherwise, the natural numbers would be closed under subtraction.

_to say that we close a set under an operation is not to say that the original set was closed under that operation_

Jon, right. I agree entirely that the original set (viz. S) was not closed under disjunction. But why then do you say ” . . .when a thing [for instance a set? For instance, S?]is closed (under some operation or function)[for instance disjunction?], we have an . . . identity between what we began with [for instance, S?] and what we end up with [for instance S’?]”. What “identity” are you referring to? It reads like your referring to (among other things) an identity between sets. But you are obviously not pointing out an identity between S and S’. I’m sure this is terminological.

Yes, identity between original set and image set is a paradigm example of closure. So, in your example, S is not closed under disjunction. If we close it under disjunction, however, we do end up with a set that is closed under disjunction (if the set is generated recursively from the original set). The generated set is then closed under disjunction because, if we apply the disjunction operation to any two members of the set, we end up with something that is already a member of that set.

I see what you want to say. S is closed under f iff. the closure of S under f is S. Of course that is different from saying that closing S under f yields S. This (the latter) is clearly false. So the set S of known propositions is closed under principle P (on this view) iff. P(members of S) = S. But then what counts as a counterexample to a proposed closure principle P? The simple answer is that we locate a case where P(members of S) =/= S. But there is no chance that this is right. It cannot simply be a matter of locating such a case since it takes little imagination to come up with a counterexample for any P. Suppose S = I know I have hands. Suppose S is closed under implication and therefore I know that I am not a BIV. Obviously I might not have so much as the concept of a BIV and might not live long enough to acquire such a concept. If I do not have the concept of a BIV and do not live long enough to acquire such a concept I do not know that I am not a BIV. The closure under implication therefore fails. And this sort of counterexample generalizes in the obvious way. There is no (non-trivial) closure principle that we cannot counterexample in this way.

Yes, closure is exactly that. And the set under question is a set of statements which becomes closed under certain epistemic principles provided there exists appropriate epistemic operators.

There is no principle under which knowledge is closed, as far as I can see (NB: I didn’t say “under which knowledge should be closed”). There might be principles under which some level of *ideal knowledge* is closed. Presumably omniscience is closed under implication. But I’m not even sure of this. For instance if for some proposition p, DefIndefp (i.e. it is definite that p is indefintiely true), then (by implication) Indefp. Since we might expect an omniscient being to know all definitely true propositions it follows by closure that omniscient being also knows some indefinite propositions. But I’m not sure even such an ideal knower as an omniscient being knows indefinite propositions.